Chapter 2.3, Problem 10E

(a) Find the (infinity norm) condition number of the matrix $A=\left[\begin{array}{cc}1& 2\\ 2& 4.001\end{array}\right]$.

(b) Let $b=\left[\begin{array}{c}3\\ 6.001\end{array}\right]$ and let $x=\left[\begin{array}{c}1\\ 1\end{array}\right]$ denote the exact solution of $Ax=b$ . Find the relative forward error, relative backward error, and error magnification factor of the approximate solution $x_a=\left[\begin{array}{c}-6000\\ 3001\end{array}\right]$

(c) Show that for any $\delta >0$, the error magnification factor of the approximate solution $x_a=\left[\begin{array}{c}1-6001\delta \\ 1+3000\delta \end{array}\right]$ is equal to the condition number of A.